TY - BOOK

T1 - Periodic Phenomena in the Theory of Large Atoms

AU - Bjerg, August Andersen

PY - 2024

Y1 - 2024

N2 - Any sufficiently advanced mathematical model for atoms should in some way reflect the periodicity known from the periodic table of the elements appearing in nature. This thesis describes, in selected models, the periodicity of large atoms. In particular, we discuss periodic phenomena in a Thomas-Fermi mean-field model for the atom. In this model, the atom is described by a self-adjoint 1-particle Schrödinger operator, and electrons see each other only through a mean-field potential coming from the semi-classical Thomas-Fermi functional density theory. This defines an atom for each atomic number Z. It is proved that the operators of this model converge towards particular self-adjoint operators (“infinite atoms”) in the strong resolvent sense, but only along certain subsequences Zn → ∞ describing the periodicity of the atoms to leading order. It is remarkable that this is a model sufficiently advanced to exhibit periodicity, while still simple enough so that one can completely describe the periodicity in the Z → ∞ limit. We further treat some more abstract mathematical theory related to this main result: Firstly, strong resolvent convergence of general self-adjoint operators is related to convergence of symmetric operators which they extend via von Neumann’s extension theory – and in particular to strong convergence of graphs of the latter. Secondly, the periodicity in the Thomas-Fermi mean-field model can be interpreted in terms of that of the scattering length (an object often studied in the physics literature) of the associated mean-field potentials. We develop a simple mathematical theory for the scattering length for a large class of real-valued potentials. As novel parts of this theory, the scattering length varies continuously as a function of the potentials, and we prove that differences in the number of negative eigenvalues of two one-dimensional Schrödinger operators can be measured by tracing the value of the scattering length along any choice of continuous curve connecting their respective potentials.

AB - Any sufficiently advanced mathematical model for atoms should in some way reflect the periodicity known from the periodic table of the elements appearing in nature. This thesis describes, in selected models, the periodicity of large atoms. In particular, we discuss periodic phenomena in a Thomas-Fermi mean-field model for the atom. In this model, the atom is described by a self-adjoint 1-particle Schrödinger operator, and electrons see each other only through a mean-field potential coming from the semi-classical Thomas-Fermi functional density theory. This defines an atom for each atomic number Z. It is proved that the operators of this model converge towards particular self-adjoint operators (“infinite atoms”) in the strong resolvent sense, but only along certain subsequences Zn → ∞ describing the periodicity of the atoms to leading order. It is remarkable that this is a model sufficiently advanced to exhibit periodicity, while still simple enough so that one can completely describe the periodicity in the Z → ∞ limit. We further treat some more abstract mathematical theory related to this main result: Firstly, strong resolvent convergence of general self-adjoint operators is related to convergence of symmetric operators which they extend via von Neumann’s extension theory – and in particular to strong convergence of graphs of the latter. Secondly, the periodicity in the Thomas-Fermi mean-field model can be interpreted in terms of that of the scattering length (an object often studied in the physics literature) of the associated mean-field potentials. We develop a simple mathematical theory for the scattering length for a large class of real-valued potentials. As novel parts of this theory, the scattering length varies continuously as a function of the potentials, and we prove that differences in the number of negative eigenvalues of two one-dimensional Schrödinger operators can be measured by tracing the value of the scattering length along any choice of continuous curve connecting their respective potentials.

M3 - Ph.D. thesis

BT - Periodic Phenomena in the Theory of Large Atoms

PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen

ER -